Stress Concentrations - University of...
Transcript of Stress Concentrations - University of...
Prof. M. E. Tuttle University of Washington
Stress Concentrations
• A “stress concentration” refers to an area in a object where stress increases over a very short distance (i.e., where a high stress gradient exists)
• Stress concentrations typically occur due to some • Stress concentrations typically occur due to some localized change in geometry (near holes, filets, corners, grooves, cracks, etc)
• These changes in geometry are often called “stress risers”
Prof. M. E. Tuttle University of Washington
Stress Concentration Near a Circular Hole
• In 1898 Ernst Kirsch (a German engineer) published a solution for the elastic stresses near a circular hole in an isotropic “infinitely
D
o
hole in an isotropic “infinitely large” thin plate (the Kirsch solution is derived in Sec 3.13 of the Shukla and Dally textbook)
• In practice, a thin plate can be considered to be “infinitely large” if the hole diameter is small compared to the in-plane plate dimensions (if a/D < ~0.005, say)
2a
o
Prof. M. E. Tuttle University of Washington
Stress Concentration Near a Circular Hole
• Stresses along the x-axis in an infinite plate predicted by the Kirsch solution:
31
32
−== o aaσσσ
0
32
2
31
2
4
4
2
2
22
==
++==
−==
xyr
oyy
oxxrr
x
a
x
a
x
a
x
a
ττ
σσσ
σσσ
θ
θθ
Figure 3.6: Distribution of σxx/σo and σyy/σo
along the x-axis
Prof. M. E. Tuttle University of Washington
Stress Concentration Near a Circular Hole
• Stresses at the edge of the hole (at x = a):
3
0
==
==
oyy
xxrr
σσσ
σσ
θθ
• The stress concentration factor for a circular hole inan infinite plate:
0
3
==
==
xyr
oyy
ττ
σσσ
θ
θθ
Figure 3.6: Distribution of σxx/σo and σyy/σo
along the x-axis
3==o
yytK
σσ
Prof. M. E. Tuttle University of Washington
Stress Concentration Near a Circular Hole
• If a/D > 0.005 then the plate is “finite” and the Kirsch solution is no longer valid
• Stress concentration
D
o
• Stress concentration factors for a circular holes infinite plates have been measuredexperimentally for a range of a/D ratios (usually usingphotoelasticity), and tabulated inthe form of curve-fits in reference handbooks …required several yearsand many contributors
o
2a
Prof. M. E. Tuttle University of Washington
Stress Concentration Near a Circular Hole
• Example: Wahl, A.M., and Beeuwkes, R., “Stress Concentration Produced by Holes and Notches”, Transaction of the ASME; Applied Mechanics, Vol 56 (11) , 1930
Prof. M. E. Tuttle University of Washington
Stress Concentration Near a Circular Hole
• Two different definitions of the stress concentration factors arein common use:
-based on the grossstress :maxσ
D
P
-based on the grossstress :
(σg remains constant as a increases)
-based on the netstress:
(σn increases as a increases)
Dt
PK g
g
yygt *
wheremax
== σσ
σ
)2(* re whe
max
aDt
PK n
n
yynt −
== σσ
σ
2a
P
Prof. M. E. Tuttle University of Washington
Stress Concentration Near a Circular Hole
• Example: from Roark’s Formulas for Stress and Strain(2002): D
o
o
2a
3
2
253.1
267.3
214.300.3
−
+
−=
D
aD
a
D
aK n
t
Prof. M. E. Tuttle University of Washington
Stress Concentration Near an Elliptical Hole
• In 1913 Charles Inglis (a British mathematician) published a solution for the elastic stresses near an elliptical hole in an isotropic an elliptical hole in an isotropic infinitely large thin plate (the Inglis solution is discussed in Sec 4.2)
• In this case the stress concentration depends on both the aspect ratio of the hole (a/b) and on the size of the plate
Prof. M. E. Tuttle University of Washington
Stress Concentration Near an Elliptical Hole
• For an infinite plate the stresses along the x-axis (i.e., for x ≥ a, y = 0) are given by:
)( )(2)(1 −= FFx ssxxσ
where:
0)(
)( )(2)(1
)(2)(1
=
+=
x
FFx
xy
ssyy
ssxx
τ
σ
ba
bambaBm
B
x
B
xs
ms
ms
ms
m
ms
mF
ms
mF ss
+−=+=−
+=
−−
−−+
−−+=
−++=
)(2
1
22
311
11
2
)1(21
2
2
2
2
22
2
)(22)(1σσ
Prof. M. E. Tuttle University of Washington
Stress Concentration Near an Elliptical Hole
• For a finite plate:
babaC
babaC
babaC
babaC
D
aC
D
aC
D
aCCK n
t
/011.4/204.5270.2
/494.4/183.5621.3
/483.2/021.0351.0
/000.2/000.0000.1
222
4
3
2
1
3
4
2
321
−+−=
+−=
−−−=
++=
+
+
+=
Prof. M. E. Tuttle University of Washington
Measuring Stress Concentrations Using Strain Gages
• In general, even the smallest of commercial resistance strain gages are too large to measure strain concentrations measure strain concentrations near stress risers:
Prof. M. E. Tuttle University of Washington
Measuring Stress Concentrations Using Strain Gages
• In general, even the smallest of commercial resistance strain gages are too large to measure strain concentrations
o
measure strain concentrations near stress risers:
…a poor experimental approach
o
Prof. M. E. Tuttle University of Washington
Measuring Stress Concentrations Using Strain Gages
• Instead, use a commercial “strip gage” and extrapolate experimental measurements to edge of stress riseredge of stress riser
Prof. M. E. Tuttle University of Washington
Lab #5: Stress and Strain Concentrations
Prof. M. E. Tuttle University of Washington
Lab #5: Stress and Strain Concentrations
Prof. M. E. Tuttle University of Washington
Corrections for Biaxial Rosettes With Differing Transverse Sensitivity Coefficients
yt
xt
myxt
ytomx
xto
xKK
KKK
−
−−−=
1
)1()1( ενενε
yxy KKK −−− )1()1( ενενyt
xt
mxyt
xtomy
yto
yKK
KKK
−
−−−=
1
)1()1( ενενε
=mymx εε , strains measured in the x- and y- directions
=yt
xt KK , Transverse sensitivity coefficients for gages
in the x- and y- directions
MM Tech-Note 509 “Errors Due to Transverse Sensitivity in Strain Gages”
Prof. M. E. Tuttle University of Washington
Lab #5: Stress and Strain Concentrations
Goals:
•To compare stress distributions measured near an •To compare stress distributions measured near an elliptical hole in a finite thin plate to those predicted for an infinite thin plate, and
•To compare the stress concentration factor measured for an elliptical hole in a finite thin plate to the value expected from a reference handbook.
Prof. M. E. Tuttle University of Washington
Lab #5: Stress and Strain Concentrations“Official Data” – Axial Strains
800
1000
1200
Axial Strains
Gage 1A
Gage 2A
-200
0
200
400
600
0 500 1000 1500 2000 2500 3000 3500 4000
Str
ain
(µε)
Applied Load (lbf)
Gage 2A
Gage 3A
Gage 4A
Gage 5A
Gage 6A
Gage 7A
Gage 8A
Gage 9A
Gage 10A
Prof. M. E. Tuttle University of Washington
Lab #5: Stress and Strain Concentrations“Official Data” – Transverse Strains
20
40
60
Transverse Strains
Gage 1T
-80
-60
-40
-20
0
20
0 500 1000 1500 2000 2500 3000 3500 4000
Str
ain
(µε)
Applied Load (lbf)
Gage 1T
Gage 2T
Gage 3T
Gage 4T
Gage 5T
Gage 6T
Gage 7T
Gage 8T
Gage 9T
Gage 10T
Prof. M. E. Tuttle University of Washington
Lab #5: Stress and Strain Concentrations
Goal 1:Compare stress distributions measured near an elliptical hole in a finite thin plate to those predicted for an infinite thin plate(Suggestion: compare normalized stresses)
6
0
1
2
3
4
5
0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45
σ yy/σ
net
Position along x-axis (in)
Inglis Solution Measured Note: as mentioned during class lecture, the measured values shown here are fictitious
Prof. M. E. Tuttle University of Washington
Lab #5: Stress and Strain Concentrations
Goal 2:To compare the stress concentration factor measured for an elliptical hole in a finite thin plate to the value expected from a reference handbook.(Suggestion: extrapolate a curve fit)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.5 1 1.5 2
Normalized axial stress
Prof. M. E. Tuttle University of Washington
Lab #5: Stress and Strain Concentrations
Goal 2:To compare the stress concentration factor measured for an elliptical hole in a finite thin plate to the value expected from a reference handbook.….fit using a 2nd-order polynomial:….fit using a 2 -order polynomial:
y = 4.84x2 - 13.269x + 10.148
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.5 1 1.5 2
Normalized axial stress
Prof. M. E. Tuttle University of Washington
Lab #5: Stress and Strain Concentrations
Goal 2:To compare the stress concentration factor measured for an elliptical hole in a finite thin plate to the value expected from a reference handbook.….fit using an exponential:….fit using an exponential:
y = 5.2287e-1.103x
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.5 1 1.5 2
Normalized axial stress
Prof. M. E. Tuttle University of Washington
Lab #5: Stress and Strain Concentrations
Goal 2:To compare the stress concentration factor measured for an elliptical hole in a finite thin plate to the value expected from a reference handbook..…fit using a power law.…fit using a power law
y = 1.7238x-1.317
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.5 1 1.5 2
Normalized Axial Stress
Prof. M. E. Tuttle University of Washington
Lab #5: Stress and Strain Concentrations
Goal 2:To compare the stress concentration factor measured for an elliptical hole in a finite thin plate to the value expected from a reference handbook.….fit using an polynomial and (1/normalized stress)….fit using an polynomial and (1/normalized stress)
y = 1.2864x3 - 5.6909x2 + 8.6649x - 3.6261
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2
1/Normalized Axial Stress